Econ 302
Assignment 6 — Solution
1.
i u(w) = 1 − e−αw , u0 (w) = αe−αw , u00 (w) = −α2 e−αw < 0 when α > 0. Therefore,
u(w) = 1 − e−αw is a risk averse utility function.
ii u(w) = ln(w). u0 (w) =
function.
1
,
w
u00 (w) = − w12 < 0, so u(w) = ln(w) is a risk averse utility
iii u(w) = wσ , u0 (w) = σwσ−1 , u00 (w) = (σ − 1)σwσ−2 . Therefore, when 0 < σ < 1, we have
u00 (w) < 0, so u(w) is risk averse; when σ > 1, we have u00 (w) > 0, so u(w) is risk loving.
When σ = 1, u(w) is risk neutral.
When σ < 0, u(w) = wσ is a decreasing function of wealth w, so it is not a “sensible”
utility function over wealth; however, since u00 (w) > 0 in this case, technically it is risk
loving.
2.
i Expected value is 0.5 × 15000 + 0.5 × 6500 = 10750 dollars.
ii Expected utility is 0.5 × ln(15000) + 0.5 × ln(6500) ≈ 9.19768.
iii ln(x) = 0.5 × ln(15000) + 0.5 × ln(6500) ⇒ x ≈ 9874.21. So the certainty equivalent is
9874.21 dollars.
iv Risk premium = 10750 − 9874.2 = 875.8 dollars.
v You will not take this project, because the certainty equivalent is below your initial
wealth of $10000, or equivalently because 0.5 × ln(15000) + 0.5 × ln(6500) < ln(10000).
3. Your expected utility from betting x dollars is p ln(w + bx) + (1 − p) ln(w − x). Therefore,
you should solve
max p ln(w + bx) + (1 − p) ln(w − x).
x
1
The first order condition is:
p
−1
b
+ (1 − p)
= 0,
w + bx
w−x
i.e.,
pb
1−p
=
,
w + bx
w−x
i.e.,
(pb + (1 − p)b)x = pbw − (1 − p)w.
Solving for x in the above equation gives the optimal bet:
x=
pb − (1 − p)
w.
b
For example, when the odds is one-to-one and the probability of winning is 60%, you should
bet 0.6−0.4
= 20% of the money in your pocket; when the odds is two-to-one and the proba1
= 40% of the money in your pocket.
bility of winning is 60%, you should bet 1.2−0.4
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4. Assume that the agent’s utility is u(w) = w.
i The first best assumes that the principal can observe the agent’s effort. Obviously the
principal pays w = 0 to the agent if he does not exerts effort.
When the agent exerts effort, the principal solves:
max pV − w subject to: w − 1 ≥ 0.
w
The solution to the above maximization problem is that the principal pays w = 1 to the
agent if he exerts effort. In this case the principal gets pV − 1 which is better than his
outside option of 0 by assumption.
ii For the second best, to induce effort by the agent, the principal solves:
max pV −(pw1 +(1−p)w0 ) subject to: pw1 +(1−p)w0 −1 ≥ w0 and pw1 +(1−p)w0 −1 ≥ 0,
w0 ,w1
where w1 is the agent’s wage when the project is a success, and w0 is the agent’s wage
when the project is a failure.
For a solution of the above maximization problem, we must have the IR constraint bind:
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pw1 + (1 − p)w0 − 1 = 0, for otherwise we can decrease w0 a little bit, which increases the
principal’s profit without sacrificing IR or the IC constraint of pw1 − 1 ≥ pw0 . But this
means that at a solution of the maximization problem we must have pw1 + (1 − p)w0 = 1,
so the principal once again gets pV − 1. In other words, when the agent is risk neutral,
the principal’s second best gives the same profit as his first best.
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